Evaluate the expression $ \frac{\sqrt{-4}}{\sqrt{-4} - 2} $ and write the result in the form \( a + bi...

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Evaluate the expression $ \frac{\sqrt{-4}}{\sqrt{-4} - 2} $ and write the result in the form $ a + bi $. The real number $ a $ equals The real number $ b $ equals

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First, simplify $ \sqrt{-4} $ as $ 2i $, where $ i $ is the imaginary unit.Then, the expression becomes:$$ \frac{2i}{2i - 2} $$Factor out 2 from the denominator:$$ \frac{2i}{2(i - 1)} = \frac{i}{i - 1} $$Multiply the numerator and the denominator by the conjugate of the denominator $ i + 1 $:$$ \frac{i(i + 1)}{(i - 1)(i + 1)} = \frac{i^2 + i}{i^2 - 1} $$Since $ i^2 = -1 $, this simplifies to:$$ \frac{-1 + i}{-1 - 1} = \frac{-1 + i}{-2} $$Divide each term by $-2$:$$ \frac{1}{2} - \frac{i}{2} $$Thus, $ a = \frac{1}{2} $ and $ b = -\frac{1}{2} $.

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Evaluate the expression \( \frac{\sqrt{-4}}{\sqrt{-4} - 2} \) and write the result in the form \( a + bi...

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